I am always curious where original copies of historical documents are kept, and if they are kept intact, are they sealed or does the public have access to them at some level (e-copy?).
Yes, they certainly do, and yes, they are available on unrestricted access, and open to historians.
EIC records covering the period 1599-1858 are mostly held in the India Office collection at the British Library, alongside about 300 private collections assembled by EIC employees. There is an incredible profusion of material – for one of my books I had to go through three classes of record for around 25 years, and calculated there was the daunting total of about 60,000 A1-size pages of manuscript in this small corner of the larger collection alone.
Dutch VOC records, covering the period 1602-1795 are equally vast and are mostly held in the Algemeen RijksArchief [General State Archive] in the Hague. As is the case with the EIC, however, some classes of records (many duplicates) were left in situ, so there are also holdings in Dehli and in Djakarta, Colombo and Cape Town. In total, the VOC papers total about 25m pages.
The EIC papers are in the process of partial digitisation by Adam Matthew, but the great majority of material in both collections can only be accessed by visiting the archives in person.