Is it possible to enter Storm Shield Home Base without actually getting a mission?
Sure, with the caveat that you can't do that while you're in a party (unless you are the party leader).
Just select "Map", go to the world section you want and click on the green "pyramid" mission icon in your homebase shield. It will select your homebase as a "mission" (it will tell you your mission is "HOMEBASE STORM SHIELD Stonewood" in the top left corner for example), and as soon as you click on the button "Select" on the bottom right it will take you and possibly the rest of your party to your homebase after searching for a server, without having any combat action or objectives there.
While you're in there you are free to collect any resources around, build as long as you like.