If pay isn't important, you could look into volunteering thru GISCorps - Home or maybe in something more local (school perhaps?) If it is for non-profit and no commercial use, you can look at getting the software thru something like Esri's program for nonprofit organizations or ArcGIS for Personal Use Program | ArcGIS for Desktop Advanced for Personal Use
If you are really enthusiastic and want to dive deeper into the development and app side of things, you can look at the ArcGIS for Developers packages....but those can get a bit pricey, but maybe they have a discount thru your school.
re: paid internships remote...I don't know that there would be a generic answer for that since each organization would be set up differently as to whether they would allow it. You may want to check around some of the web sites for the type of company/discipline that you plan to follow with your GIS career. They may have something on their sites...or maybe you can talk to someone in their HR or GIS shop for some tips.
edit: Micah beat me to the GISCorps