Is Unity3D capable of handling a massively multiplayer online real-time 3D shooter? (i will be doing the networking by myself)
For the client, yes.
For the server, I strongly think not, though others may disagree.
You can certainly run mmo-s with unity, though you will need to make sure everything is very optimized, and remember that unity was not designed for mmo-s so there will be some limitations.
We have a tutorial on networking here if you need some help:
http://cgcookie.com/unity/2011/12/20/introduction-to-networking-in-unity/
Feel free to let us know if you get stuck at all as it is a difficult process 
I’m currently struggling to get Unity into MMORPG world both as client and server (in conjunction with uLink). It’s sometimes very hard to synchronize everything for many connected player - we’are currently trying to introduce mecanim anim system into mmo and we’ve succeded. However there are many areas where you can encounter problems with Unity.