Is a2x8x10' enough support as a ridge beam for a 14'x10' shed roof?
I assume from your description that your are stick framing (ie not using trusses, and not with any ceiling joists.

The short answer to your question is "maybe", but probably not. The answer to the question depends on the expected wind and snow loads in your area, the species of the wood you plan on using, whether it's normal dimensional lumber, or a laminated beam. 

My best advice is to find the permitting authority for your area (typically either the city or building department), and go in and talk with an inspector. They can give you a much more definitive answer than you can get from us, and you can be more confident in it. If you are in an area where that is difficult, the best answer I can give you is that a single 2x10 may work if you have very light loading, but I would be more comfortable with a doubled 2x10 instead. I tend towards overbuilding in cases like this, and the cost of a doubled 2x10 in cost and effort it minimal. 

Note that with a ridge beam, you need to carry the load in an uninterrupted path from the beam all the way down to the footings. A 4x4 should be fine for that.

Hope that helps.