Considering that there were very few weapon types, I believe we can reasonably protect against all of them. The gear would have to be something resistant to piercing and cuts (swords, spears, arrows) with a good amount of padding underneath to defeat mauls and other blunt weapons. This is a jack of all trades -version of such armor and should offer some protection from each weapon type.
Let's say a protective outer layer made of layered kevlar or dyneema, with polyurethane coating. (This setup is common in cut-resistant items, such as gloves designed to protect from knife cuts.) Layer with cushioning materials underneath (and perhaps a couple of cushioning layers in the outer part as well), and you've got yourself a pretty solid set of protective clothing. As an added benefit, it should be fairly light compared to what was used in medieval combat.
This armor is, however, still susceptible to fire and will not absorb all the energy from a very determined flail-swinging muscleman or a lancer, among other things. It will, however, provide decent protection in a melee, turning hits that would outright kill an unarmored person into mere bruises or relatively minor injuries. As an added defense against crossbow bolts or arrows one could add a removable ceramic plate to the front part of the armor, although this would add considerable weight.
Edit: Another reply correctly pointed out that ceramic armor is designed to absorb the energy by breaking, making it only usable against a single or at best two or three hits. In this, however, the medieval armorsmiths had the right idea when they created scale mail. There have been concepts thrown around of a scale mail made out of ceramic armor, which could be a very decent solution against crossbow bolts and arrows. It would, however, add considerable weight, so the usefulness of such a design remains debatable and depends entirely on how thick the scales would have to be in order to be effective.