SHORT ANSWER
In antiquity, the Scythians (Eurasian nomads) and the Sarmatians (nomads of Iranian origin who moved westwards, gradually overwhelming the Scythians) had significant numbers of female warriors. Estimates based on archaeological discoveries range from 15% to over 30% of women who were warriors. The precise role of these fighters remains unclear, as do the reasons why these related cultures had female warriors while other, similar, cultures apparently did not. Note:'Scythians' is sometimes used to include 'Sarmatians' while at other times it refers only to the people west of the Don River. This, as Wikipedia notes, has led to a fair amount of confusion.
MAIN ANSWER
Archaeological finds, with the aid of science, over the last 25 years have provided an increasing amount evidence that Scythian women and those of their eastern kin the Sarmatians (among whom the Sauromatae are most frequently mentioned) fought in significant numbers, and that the writings of Herodotus and other ancient historians on this subject have at least some basis in fact. Earlier finds have also been reassessed as it was previously assumed that any grave containing weapons belonged to a male. Deborah Levine Gera, Professor of Classics at the Hebrew University of Jerusalem, in Warrior Women: The Anonymous Tractatus De Mulieribus asserts:
The presence of...Scythian women warriors in the pages of Herodotus
and Ctesias reflects some kind of historical reality, for there is
evidence that some Scythian women rode on horseback, used bows and
arrows, and went into battle. Archaeological remains indicate that
there were female Scythian warriors, chiefly - but not solely - among
the Sauromatae,..

This Sarmatian female warrior tomb "was found with more than 100 arrowheads, a horse harness, a collection of knives and a sword". Source: ZME Science
In The Scythians 700-300 BC by the archaeologist Dr. E. V. Cernenko, the author asserts:
Almost the whole of the adult population of Scythia, including a large
number of the womenfolk, fought on campaign.
What evidence is there for the above, other than Herodotus? For the archaeological evidence, the Smithsonian article The Amazon Women: Is There Any Truth Behind the Myth? from 2014 relates the following from the early 1990s:
...a joint U.S.-Russian team of archaeologists made an extraordinary
discovery while excavating 2,000-year-old burial mounds—known as
kurgans...outside Pokrovka...near the Kazakhstan border. There, they found over 150
graves belonging to the Sauromatians and their descendants, the
Sarmatians. Among the burials of “ordinary women,”... There were
graves of warrior women who had been buried with their weapons. One
young female, bowlegged from constant riding, lay with an iron dagger
on her left side and a quiver containing 40 bronze-tipped arrows on
her right. The skeleton of another female still had a bent arrowhead
embedded in the cavity....On average, the weapon-bearing females
measured 5 feet 6 inches, making them preternaturally tall for their
time.
The last sentence is interesting as it deals with the perception that women are physically much less well equipped to fight compared to men; these women seem to have been an exception. It is also worth considering that a child, male or female, brought up from a young age to ride horses and use a bow and arrow, is likely to become a formidable adversary.
The Smithsonian article continues with
In recent years, a combination of new archaeological finds and a
reappraisal of older discoveries has confirmed that Pokrovka was no
anomaly.
According to Kathryn Hinds in Scythians and Sarmatians, the Pokrovka graves were those of "ordinary people" (not royalty), and following figures give an idea of the percentage of female warriors, at least within one community:
The vast majority of the men — 94 percent — were buried with
weapons...
15 percent of the women were warriors, buried with arrowheads and
other weapons.
Other discoveries suggest a higher percentage. This article from the Irish Times seems to refer to a more recent find:
A team of archaeologists investigating 2,400-year-old burial mounds
built by the Scythian people on the upper reaches of the river Don has
found that five of 21 graves contained the bodies of young women,
accompanied by their weapons.
The article also quotes Dr Valery Gulyayev, of the Russian Institute of Archaeology:
"Usually such women are found in large kurgans, buried with the same
rituals as for men," ... "They are buried with womanly things -
mirrors of silver and bronze, necklaces of gold, glass or clay,
earrings. But alongside these they are buried with weapons - a quiver,
bow and arrows, and, often, two throwing spears.
The National Geographic article Amazon Warriors Did Indeed Fight and Die Like Men cites archaeological finds of Scythians which have been subjected to scientific testing:
Archaeologists have found skeletons buried with bows and arrows and
quivers and spears and horses. At first they assumed that anyone
buried with weapons in that region must have been a male warrior. But
with the advent of DNA testing and other bioarchaeological scientific
analysis, they've found that about one-third of all Scythian women are
buried with weapons and have war injuries just like the men. The women
were also buried with knives and daggers and tools.
Unfortunately, the article doesn't say which specific find the above relates to, but there is again some evidence here of 'significant numbers'. The New Yorker article The Real Amazons, citing Adrienne Mayor, research scholar in the Classics Department at Stanford University, says:
...in some archaeological digs in Eurasia, as many as thirty-seven per
cent of the graves contain the bones and weapons of horsewomen who
fought alongside men. (“Arrows, used for hunting and battle, are the
most common weapons buried with women, but swords, daggers, spears,
armor, shields, and sling stones are also found,” Mayor writes.)
In the light of the archaeological evidence, there has unsurprisingly been some reassessment of ancient sources. Further, it should be noted that Herodotus was not the only writer to refer to female warriors; there is also Ctesias, Hippocrates (see the passage cited here) and - as J Asia mentioned in his comment - Diodorus. They have embellished in parts and got some details badly wrong, but the archaeological evidence seems to support the assertions of a significant number of female warriors among some of the Scythians and some of their kin, perhaps most notably the Sauromatians.
Role of female warriors
The precise role of female warriors is unclear but it is most likely that they (1) defended the community while the main body of fighting men were away, and (2) were 'called up' and fought alongside men in times of great need, such as the Persian invasion under Darius I (ruled 522 to 486 BC). Their involvement in the army may well have gone beyond these, but archaeology has yet to conclusively prove this.
Reasons for significant number of female warriors
Also unclear is why these two cultures had significant numbers of female warriors while others around them apparently did not. None of the sources cited here deal with this directly, but several suggest that the prominent role women played as rulers is significant. For example, one archaeological dig found that over 70 % of the central graves (i.e. those of highest status) had female remains.
David W. Anthony, in The Horse, the Wheel and Language, notes an interesting point about the Scythians and the Yamna people "dating to 3300–2600 BC" who were there much earlier,
About 20% of Scythian - Sarmatian "warrior graves" on the lower Don
and lower Volga contained females dressed for battle as if they were
men...It is at least interesting that the frequency of adult
females in central graves under Yamnaya kurgans in the same region,
but two thousand years earlier, was about the same. Perhaps the people
of this region customarily assigned some women leadership roles that
were traditionally male.
This, though, is only the beginning of an answer, but we can also consider the prominent role played by archers on horseback where physical strength (though not unimportant) plays less of role than it would in close-quarters combat (as suggested by orangesandlemons in his comment). We should also not forget that in any society, there are always some women who are physically stronger than some men. A further point worth mentioning is that there may have been much less gender distinction in the division of labour than in other cultures; this is in evidence among modern Kazakh nomads where boys and girls compete directly "in riding exercises and games" (DNA testing showed that one girl had the same common ancestor as a 5th or 4th century BC female warrior buried at Pokrovka).
All-female society and Thracian female warriors
Two further points worth commenting on are first, the claims (particularly in Herodotus) concerning an all-female warrior society and second, claims that there were significant numbers of female Thracian warriors. On the former,
As yet, Davis-Kimball (2002) notes that there is no archeological
evidence linking all the storied versions (e.g., no excavated
settlement suggests that women warriors lived in societies without
men).
On the latter, there is a lack of sufficient evidence, as demonstrated in Fingerprinting the Iron Age (Nicolae Popa, Simon Stoddart, eds.). However, Women in Antiquity (Stephanie Lynn Budin, Jean Macintosh Turfa, eds.) does cite one interesting discovery.
Other sources:
Hamid Wahed Alikuzai, A Concise History of Afghanistan in 25 Volumes, Volume 14
Jeannine Davis-Kimball, Warrior Women of Eurasia (abstract) in Archaeology, A publication of the Archaeological Institute of America
The Cambridge History of Early Inner Asia, Volume 1