Is there an accepted maximum allowed size for HTTP headers? If so, what is it? If not, is this something that's server specific or is the accepted standard to allow headers of any size?
No, HTTP does not define any limit. However most web servers do limit size of headers they accept. For example in Apache default limit is 8KB, in IIS it's 16K. Server will return 413 Entity Too Large error if headers size exceeds that limit.
413 Entity Too Large
Related question: How big can a user agent string get?