The repositories (packages.linuxmint.com) are currently down, giving 403 errors or not responding at all.
Sorry for the inconvenience. We’re aware of it and we’re trying to fix them as quick as possible.
Affected parts of the desktop: APT, mintInstall and Synaptic can only install from Ubuntu or 3rd party repositories. MintUpdate cannot work at all.
Update (7th July, 4:38pm GMT0): I made a temporary fix. The repository is working at the moment but expect some more downtime later when this gets properly fixed by Michael (d00p).