Are GFCI protected outlets a valid solution for ungrounded outlets?

Hello, after reading about how many older homes have ungrounded outlets and how even three prong outlets might not necessarily be grounded, I got a receptacle tester for my 1955 home.  And turns out that just about every upstairs outlet is in fact ungrounded (prior owners finished the basement about 10 yrs ago and so those outlets are all grounded along with the upstairs kitchen which was remodeled).  These ungrounded outlets are all in our bedrooms as well so quite concerning.  In looking into what it would take to fix, this work would be quite expensive and invasive (going through the walls and such) and so I am curious about the viability of using GFCI protect outlets in lieu of fully grounding these outlets.  It appears that these would solve the immediate fire danger of ungrounded outlets but I am curious whether that addresses the EMF coming off of these outlets as well.  Any info appreciated here.  I would like to avoid having to spend a few thousand dollars to address this, but obviously for the sake of my family I will do what it is required to fully address the issue (both fire safety and EMF).