Option A: Textile Armor
Gambison was a historical style of armor made from layers of linen sewn together. A good gambison can stop nearly all historical arrows fired even at relatively close range as well as duller bladed weapons. Most arrows just bounce off, but when they do penetrate (like when you fire an English war bow or crossbow at close range), it typically askews the path of the arrow as it penetrates each successive layer until it is turned to travel between the layers of cloth instead of through it.
That said, melee weapons are less prone to loose their alignment on a hit because they are braced in your hand so spears, war picks, thrusting swords, and particularly sharp curved swords are all fairly good at penetrating it.
Option B: A Hide and Wicker Laminate Shield
A shield does not need nearly as much stopping power as armor to save you from an arrow. Whereas armor needs to completely stop a weapon's penetration to keep you safe, shields are held away from the body; so, even when an arrow head goes through a shield, the friction with the shaft can still stop it before it harms the user.
The Persian Empire took advantage of this fact when they designed their shields. Mobility and archery were major aspects of their military doctrine; so, instead of making heavy shields that would slow them down but block almost anything like most other ancient civilizations, they made these very light weight tower shields that could screen their entire body, but just from arrows. The shields could be easily hacked through with a sharp sword or over penetrated with a spear thrust making them almost useless in melee (as the Greeks proved on more than one occasion), but against arrows, they were a very effective countermeasure for their weight.
This same principle can also be applied to situations where you just need a "thing" to block an arrow. So, a thin wall or door could also apply as long as you are not standing directly against it.
Option C: Plate Armor
If you slightly shift your thinking to just being about overcoming the armor in melee, but not with any arrow, then plate armor may be another solution. Tests of historical recreations of plate-mail against historical recreations of all sorts of weapons pretty consistently show that cutting or piercing it was practically impossible. Killing a plate armored warrior with a bow generally required a lucky shot to enter through an eye socket or exposed joint. By the 14th century, plate armor had become so well made that it was virtually impossible to get around with an arrow.
That said, a knight could still be killed through his armor by blunt force trauma. Front-heavy weapons like maces, axes, and war-hammers could kill a knight without actually penetrating the armor at all. This was even more true in the ancient era before the wide-scale use of arming jackets to cushion the blow. Daggers were also a popular option because their shorter blades gave one enough tip control to aim it into the very narrow eye slits which an arrow or sword would almost never be lucky enough to get into.