Is the data gathered from telescopes "open source"? If so, is there any website that group it all?
Most general purpose observatories release the data taken on their facilities after the expiration of the proprietary period (this is the time, typically 12-18 months, where the data is only available to the proposers for the telescope time so they can work on it without being "scooped"). "Focused telescopes" in the sense of those that are performing surveys or looking for specific types of objects, tend not to release the raw data, instead releasing processed data or catalogs.
Some examples of data archives that provide raw or processed data include:
- the NOAO Science Archive: provides access to 40 telescope+instrument combination from the US National Optical Astronomy Observatory
- the ESO Science Archive Facility: access to raw and processed data from the four 8-m VLT telescopes plus the VLTI and the smaller (1-3.6m) telescopes on La Silla, Chile
- the MAST archive at StSci: contains data from Hubble, Kepler and K2, TESS, SwiftUVOT, XMM and many others
- the ING archive: for the Isaac Newton Group of telescopes of La Palma (1.0m JKT, 2.5m INT, 4.2m WHT telescopes)