@Nathan Gesner it seems like all this stuff could be a lease violation in several ways (depending on how your lease is set up). I think your best option is to have the tenant get a storage unit, so that they can greatly reduce the amount of stuff they're storing at your property.
A few things to consider:
1. leases often have sections describing appropriate (and inappropriate) use of the unit. Having stuff stacked to the ceiling and spilling out all over the yard seems like an inappropriate use of the property. Is it possible to add a section to your lease that specifies how much personal belongings someone can store in the unit? ...perhaps the lease could state that if the tenant exceeds that threshold, then they're required to get a storage unit? ...I have no idea whether it would be legal to have this in a lease, so you'd want to check with an attorney... but if it is legal, then that seems like a reasonable solution...
2. leases usually have sections describing cleanliness requirements, and this mess would probably be a violation...as others have mentioned, when the tenant leaves, they'll probably also leave you with a giant cleaning bill if you don't take action first...
3. Perhaps most importantly, all this stuff is probably a fire hazard. Presumably, a lot of it is combustible. Plus, all this stuff lying around could easily impede ingress/egress. I would assume that the fire marshall would not be pleased if they saw this mess...
Lastly, just from a common-sense perspective, if someone has so much stuff that it's stacked to the ceiling and spilling out all over the yard, then that's an unreasonable amount of stuff, and it's time for the tenant to get a storage unit (or rent a larger place).
Think of it this way: If a tenant owned ten cars, you wouldn't let them park all over the yard and take up every parking space. They'd be required to pay for extra parking spaces, or buy parking spaces at some other place. If a tenant wants to have this much stuff, then they need to rent a place that's large enough to hold all the stuff, or pay for a storage unit. Simple as that.
So, ultimately my suggestion is to look into whether you can design your lease such that you're able to prevent and address hoarding situations. It seems reasonable to me to say to the tenant: "This is a problem because of X, Y, Z. You need to either get a storage unit, or rent a larger space" (of course, be aware that if you rent them a larger unit, they'll likely fill it with more stuff!).
Good luck out there!