First of all – spread the word among your family/freinds that you're taking up potted gardening. You'd be amazed at how many people are willing to propogate off hteir own plants or keep an eye out for large pots for you.
Then, now this may sound odd – but you can “mock” grass by installing fake turf. This can be professionally done for around $.25 sq foot, which can be very expensive
but if it's only a rental, use a lesser grade material purchased at a hardware store.
Lay a bed of pea gravel (the pros use crushed granite) for drainage. Lay the “grass”
and stake it down every 8″ or so. Be sure to poke holes where any liquid can drain
through into the ground.
Now the reason I said to start the pots, is because you're only going to put
the “grass” where you need it, not all over the yard. This way, you can create
walkways, seating areas, which might be deisrable with a BBQ, and even a sandpit
or two where your animals may prefer to mess. Anywhere you have a “bald” spot,
add a pot. If finances don't alow for the entire yard to be compelted at once, put up
some of that reed fencing to hid parts until you can get to it. Or fence in the portion you want to enjoy for yourself right out the backdoor, and hide the rest of the
ugliness.
Hope this is a reasonable solution for you.