Depends on the type of archer in question. The higher grades of professional archers, like Genoese crossbow companies or English retinue men, could afford to be equipped well enough that they were a force to be reckoned with in hand-to-hand combat. Standard English tactics of the day had the archers come forward to help outflank the enemy, which would not have been possible if none of them were armored. Poor men might only have been able to acquire padded shirts, but the richest archers (retinue men, the captains and lieutenants of the archer companies) could have far better, like mail shirts. However, even the richest archers would not be equipped as heavily as knights and men-at-arms were. Most sources, when describing the English archers participating in hand-to-hand combat, emphasize their mobility. They didn't need to be heavily armored, because the men-at-arms would be handling the brunt of combat. They needed to be light and mobile to outflank the enemy and move from position to position, filling in gaps and preventing isolated English units from being engulfed by numerically superior opponents.