If a person studies and/or does experiments at his/her own home or laboratory because of some reasons, is he/she still allowed to compete with others to get a prize like Nobel Prize, Field Medal, etc?
More precisely, from the beginning he/she never gets educations from formal institutions (schools, colleges, universities).
A person is nominated for a Nobel Prize by someone familiar with his work, and the Nobel Prize committee judges it for its originality, depth, and service to mankind.
It's barely possible for someone to make a highly original contribution to a field outside of the usual academic circles. In the unlikely event that this occurs, the Nobel Prize committee will consider it on its own merits.
For instance, the Nobel Prize was awarded to a new finance concept called microlending, developed by a banker, not a professor. Admittedly, it was the Nobel Peace prize, but it could have been awarded as the Economics prize.