OSM API doesn't allow to download areas greater than few sq km. Try to use downloading using overpassAPI - it's in File menu, but this also doesn't allow you to download all planet. Keep in mind that more data = less performance in JOSM.
Try some download sources listed at https://wiki.openstreetmap.org/wiki/Planet.osm and make use of tools like "osmconvert" and "osmfilter" (see OSM-Wiki) to get raw OSM data for bigger areas.
The main API is essentially an editing API only. It should not be used to perform bulk downloads.
See the country and area extracts. If they are still too large for you then see OSM file formats which has some notes about processing these files and splitting them into different geographical regions, e.g. by using osmium.