If I shared a Marketplace policy with more than one person, do I need to enter all the people's social security number?

Yes, full information requested will be need to be entered in the program.

The program only allowed me to enter only one other person. I have three people in the policy.

I just tried in my practice program and had no problem. Delete the form and try again.
Here are the steps for deleting unwanted forms in TurboTax Online:
Open your return in TurboTax. (To do this, sign in to TurboTax, and select the Take me to my return button.)
In the left side bar, select Tax Tools> Tools.
In the pop-up window Tool Center, choose Delete a form.
Select Delete next to the form/schedule/worksheet and follow the onscreen instructions.