Design and Appearance Edit
Room 303, like the other rooms within the hotel, had a vintage look. The rooms were small, square-shaped and had a door to what presumably was an en-suite bathroom.
However, this room, unlike the others, had a chair and a long table that was used by members of the Resistance to place their computers on whenever they required space to research their bluepill target potentials. The room, with its vintage style, was also relatively clean and less dusty because Resistance members used the room regularly for business. The room was used for reaching an exit, where redpills could safely exit from the Matrix without any bluepills, Agents, or others seeing them.
The number "303" appears to be an allusion to the character Trinity, whose name in Christian theology defines the three-in-one nature of the single Christian God: Father, Son, and Holy Spirit. This allusion is supported by Trinity kissing the lifeless Neo, whose life functions return as Neo transfigures into the One after exiting the room.
The number is also used as a parallel to Room 101, Neo's apartment, also seen near the start of the film. 101, a number used to describe beginners' tutorials, is contrasted with 303 for Trinity (whose name means "three").
Some also interpret a symbolic spiritual significance to the room. The Matrix begins with Trinity within the room and ends with the One exiting the room and resurrecting from the dead. The parallel between the character of Trinity and Christianity's Holy Trinity, emerges from the room as those who enter the room receives spiritual power and guidance. Neo resurrects from the dead, Trinity receives help from Morpheus via phone to find an exit and escape safely into the Real World.
The number 303 is also used in the status code "HTTP 303", which is a way to redirect web applications to a new URI (Uniform Resource Identifier). Bluepills can be 'redirected' or 'transferred' from the Matrix to the real world via a telephone call. Edit
303 could also be a reference to "Roland TB-303" a bass synthesizer, which is often used in electronic dance music genre. The track "Mind Fields" by The Prodigy on The Matrix' soundtrack album, used a Roland TB-303.