You would usually need a degree, followed by a postgraduate qualification in conservation. Degrees in conservation are available, but other subjects like fine art, ceramics and glass, textiles or science (particularly chemistry) are also useful.
Use the course search on this site to look for suitable courses and check directly with course providers for their entry requirements.
For most courses you would need to have work experience in a museum, or other historic site.
Large museums usually have lots of requests for volunteering, so it may be best to try smaller local museums. You can find addresses in The Museums Yearbook, which is published each year by the Museums Association. You should be able to find the book in reference libraries.
See the Museums Association website for advice on finding opportunities for volunteering.