If you hold a UK passport, you're entitled to spend as long as you like in any one of the 24 other EU countries.
You can take up permanent residence in any of those countries and you can also work in those countries. However, if you wish to take up residency of another EU country, you may be required to obtain a 'residency permit'. You should do so within 3 months of entering the country. Details are here:
http://www.consulfrance-londres.org/article.ph p3?id_article=379&PHPSESSID=12294e5e3caf116535 c92cd4b306946a
Your question implies that you don't intend to take up residency within France but if you stay at the same address for a prolonged period it might still be best to obtain a residency permit. However, if you simply intend to tour France (and/or other EU countries) for a prolonged period then you won't need a residency permit.
Your question actually asks how long you can stay out of the UK. As far as the UK authorities are concerned (irrespective of whether your destination is within the EU or elsewhere) the answer has to be "for ever". There is nothing in UK law which prohibits anyone from leaving the country. (Theoretically, you don't even need a passport. A passport is only required to enter another country, not to leave the UK).
Chris