Technically, it's not the virus(1) per se that hides in the video but specially crafted media data. The data can exploit either a vulnerability that will cause the media player to misbehave (e.g. for vlc or media player) or loose rules about extented media content (Note: link to an article that advertises a product - I'm not affiliated with it, though).
[...] can the video itself still be playable and function normally [...]
It depends on the attack method. Exploiting a vulnerability of the media player usually disrupts the normal flow of the player - so you'll be able to notice it. If you are just redirected to a "codec download" and tricked to execute malware, then you won't notice anything (at least the first time, because if you keep being redirected then obviously something's wrong)
[...] are there any limits to what a such virus can do [...]
Depends on the account privileges with which the malware infects your system. If you are infected when being a non-privileged user then there are some limits and contraints on what the malware can do. If, however, the malware infects the system when you are a privileged user (e.g. admin) or it has a way to escalate its privileges (e.g. kernel exploit) then all bets are off.
(1) You can, however, trick an MS-Windows user into thinking that an executable malicious program ending in e.g. .mp4.exe is just a video and execute the malware directly, but this is a different discussion