Is the scarlet letter 'A' a real historical thing?

tl; dr

• There were certainly laws requiring that adulterers had to wear the letter 'A' stitched upon their garments in the late seventeenth century.
• The letter was not required to be scarlet - just a different colour from the clothes themselves, so it would stand out.
• The law also applied to both men and women.

However, during the time in which Nathaniel Hawthorne's novel The Scarlet Letter is (1642 to 1649), the penalty for adultery in Massachusetts was death.

Wearing the letter 'A'

By the late seventeenth century in Massachusetts there were certainly laws requiring that convicted adulterers be whipped and then have to wear the letter 'A' stitched upon their garments for the rest of their lives. There was no requirement for the letter 'A' to be scarlet. It simply had to be a different colour from the clothes themselves, so that it would stand out.

The law applied to both men and women.

For example, a 1694 "Act against Adultery and Polygamie", passed in Massachusetts stated that:

And if any man shall commit Adultery, the Man and Woman that shall be Convicted of such Crime before Their Majesties Justices of Assize and general Goal delivery, shall be set upon the Gallows by the space of an Hour, with a Rope about their Neck, and the other end cast over the Gallows : And in the way from thence to the Common Goal, shall be Severely every Whip't, not exceeding Forty Stripes each: Also every Person and Persons so Offending, shall for ever after wear a Capital A of two Inches long and proportionable bigness, cut out in Cloath of a contrary Colour to their Cloaths, and Sewed upon their upper Garments, on the out-side of their Arm, or on their Back, in open view. And if any Person or Persons, having been Convicted and Sentenced for such Offence, shall at any time be found without their Letter so worn, during their abode in this Province; they shall by Warrant from a Justice of Peace, be forthwith apprehended and ordered to be Publickly Whip'd, not exceeding Fifteen Stripes; and so from time to time toties quoties.

• (my emphasis)

["toties quoties" is a Latin phrase meaning 'as often as the occasion arises']

The punishment for adultery in the time The Scarlet Letter was set

However, as you note, the book, The Scarlet Letter is set rather earlier - between the years 1642 to 1649. At that time, the laws in Massachusetts were not nearly so lenient as to allow offenders to get away with a simple whipping and then wearing a letter stitched to their clothing as a badge of shame.

Adultery was a capital crime.

The Massachusetts Body of Liberties of 1641 was,

"... the first legal code established by European colonists in New England."

It declared adultery to be a capital offence, in accordance with the biblical law of Leviticus:

"If any person committeth Adultery with a maried or espoused wife, the Adulterer and Adulteresse shall surely be put to death."

• The colonial laws of Massachusetts : reprinted from the edition of 1660, with the supplements to 1672 : containing also, the Body of Liberties of 1641 pp54-55

(As you can see, the relevant passages from Leviticus and Deuteronomy are also cited in the text)

Adultery persisted as a Capital Crime at least as late as 1660 where it again appears in the Colonial laws of Massachusetts:

If any person commit ADULTERY with a married or espoused wife, the Adulterer and Adulteresie shall surely be putt to death. Levit. 20.19. & 18.20. Deut. 22. 23.27

• The colonial laws of Massachusetts : reprinted from the edition of 1660, with the supplements to 1672 : containing also, the Body of Liberties of 1641 p128

If you are interested, a recent (2013) article titled On the Trail of the Scarlet AD, by Joel S. Berson, published in the Nathaniel Hawthorne Review (Vol. 39, No. 1 (Spring 2013), pp. 133-154, Penn State University Press) presents the results of Berson's fascinating investigation into the colonial laws and legal references that served as source material for The Scarlet Letter.

I haven't been able to find an online copy that isn't behind a paywall, but you may be able to get a copy through a local library.