How compatible is Libra woman and Virgo man?

If you take Astrology seriously you shouldn't base anything important on just one factor such as sun-sign -- there are so many other factors to take into account. You can compare birth-charts at a site like astro.com. Even if you don't have the time and place of birth, you can still get quite a lot from the full chart, including Venus and Mars, and probably the Moon.
And if you don't take astrology seriously, then it really doesn't matter one way or the other!

Traditionally signs that are right next to each other like Virgo and Libra don't have a lot in common, but it really means nothing until you see where the moon is, the rising sign and so on.