There are many wonderful sources for full-length, legal, video of theatrical productions. Some are free. Some you have to pay for. Below are a few examples.
PBS provides access to many of their productions, some in perpetuity, others have experation dates.
The Space is a project that began in the spring of 2012, as a part of the Cultural Olympiad to accompany the London Olympics. They posted new video covering all of the arts almost daily, some days it was entire productions streaming live from The Globe Theatre, other days it was old, nearly-lost films by known directors, all manner of material. They were 'reimagined' a couple of years later and they now describe themselves as
"The Space brings arts organisations, artists and technology together to make great art and reach new audiences. We do this by commissioning projects, building digital skills, and helping organisations and venues reach more people using digital media."
The Space has recently produced, in commemoration of the 350th anniversary, an amazing recreation of the Great Fire of 1666 in which the majority of the city of London burned down. You can watch that here and if you want to watch it without the commentary, you can do that here.
Europeana - enables people to explore the digital resources of Europe's museums, libraries, archives and audio-visual collections. It promotes discovery and networking opportunities in a multilingual space where users can engage, share in and be inspired by the rich diversity of Europe's cultural and scientific heritage.
UWM Subscritions (i.e. not free)
Digital Theatre Plus - Provides streaming access to digitally captured 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. It also includes archival captures of Broadway productions and opera performances by the Spanish company Gran Teatre del Liceu. A wide selections of behind-the-scenes documentaries and teaching/learning resources to facilitate a deeper understanding of the works being presented are also available.
Films on Demand - Films on Demand is a streaming video collection of over 25,000 high-quality digital programs and video clips, covering topics in the performing arts, humanities, social sciences, sciences, business and economics, and health.
Kanopy - Provides a streaming platform and a broad catalog of over 26,000 streaming videos to choose from, representing more than 800 leading producers. It includes also over 400 Criterion Collection films.
SWANK - 1,000 selected feature films and documentaries produced by major Hollywood and independent studios. Films are licensed for individual student, staff and classroom use but do not include public performance rights.