Does anyone have experience in Sweden?
I have applied for positions there upon invitation.
Is there a website that lists academic or non-profit sector jobs?
To the best of my knowledge, there is not a single site. There are many sites, each university has a site, and some disciplines (like math, philosophy) have sites. I have not seen a museum oriented site.
http://www.state.gov/m/dghr/flo/rsrcs/pubs/4510.htm lists many sites. The government ministry of Education, Research and Culture is the one that oversees education and museums, you could give them a ring and see if they have information.
Does anyone know what the job market is like for museums or academia.
In academia, it is tight like the U.K., they prefer to hire people from Scandanavia, like the U.K. prefers people from the U.K. That said, it is about networks, like all academic job hunts. Once you are there, you should go talk to people, become friends and colleagues, and then you will be invited to apply. If you apply blind, you'll probably have little chance. From here, I would start sending letters of introduction to local museums and university departments, letting them know you are going to be visiting their area and perhaps staying a while, and are looking to build working relationships, or somesuch. read Phil Agre's Networking on the Network, it is online.