Is it possible to enter the institute in fallout 4 without doing the Main Mission Quests or enter through another factions like the BOS or Railroad? Can you just enter their base like you can with BOS and the Railroad without following any Main Mission quests or is it only possible to enter through the main mission?

Main mission.

The main quest is designed so you have to make contact with every faction and befriend them just a little.

Min mission only and beware: before you enter the institute you have to pick one of the three already known factions to help you.....once you make that choice the end game can only be played with that faction or the institute, not with the other two you didn't pick to help you.