There are many wonderful sources for full-length, legal, free video of theatrical productions. Many theatre companies also have YouTube channels or paid streaming services.
streaming access to British theatrical, dance, opera, and symphonic performances from houses such as the The Globe Theatre of London, The Royal Shakespeare Company, the Young Vic, The Liverpool Everyman Theatre, & The Royal Opera House at Covent Garden. Behind the scenes documentaries and educational materials also included.
National Theatre Collection brings the stage to life through access to high definition streaming video of world-class productions and unique archival material offering significant insight into theatre and performance studies.
The Space is a project that began in the spring of 2012, as a part of the Cultural Olympiad to accompany the London Olympics. It was re-opened as a support to artists and organizations, among other things, who are exploring the use of digital transmission of their content. There are some remarkable full productions available to watch.
Internet Archive contains mostly older videos that have been digitized and can be temporarily "borrowed" online. To find a play, search the title and use the "movies" format filter on the left.