Would a Server be Accessible both from Earth and Mars?
Depends on how you use it. There is no reason all of your favorite websites and services can’t remain accessible via a process known as “mirroring.” Which is a process in which local (Martian) servers will maintain a copy of all of the data on the Earth-based servers. While it may take hours for Netflix’s martian server to download a movie, it will appear to you to begin streaming instantaneously since you are actually streaming from a locally stored copy. This is just an extension of what is already being practiced. Per example, if you download a Netflix movie, you are not actually downloading the movie from the main Netflix server, but from the one closest to you, which is an identical replica.
Two-way communication, sending and receiving information/messages, will still experience delays between 4 and 24 minutes, depending on how far Mars is from Earth, where they are in their respective orbits.