In English, we have lots of words that come from the Latin tenere, meaning "to hold, or to keep." This tenere family includes words like "contain," "attend," "entertain," and "tendency," as well as the ones we're checking out right now: "tenable" and "untenable."
Something tenable is good enough, strong enough, or steady enough to work or to continue. Tenable things are doable, workable: they're able to function, able to stay the same, or able to keep going—usually because they're logical, reasonable, or practical.
In other words, something tenable is likely to hold together, or at least unlikely to fall apart.
A tenable plan or a tenable solution is likely to work. On the show Firefly, Kaylee explains to Simon that they have a tenable plan. She says: "It's a train heist. See, we fly over the train car. The captain and Zoe sneak in, we lower Jayne onto the car, and they bundle up the booty, and we haul 'em all back up. Easy as lyin'." Kaylee believes it's a tenable plan: that they'll accomplish it quickly and get away with the crime easily.
Now, an untenable plan or an untenable solution is likely to fail or fall apart. Kaylee is wrong: the train heist is untenable. Jayne gets injured, the captain and Zoe get caught, and in the end, nobody steals anything.
Often, it's plans and solutions that we describe as tenable or untenable, but we also talk about tenable and untenable situations. If you're in a tenable situation, everything is fine: you can handle it, and you can continue living your life in that situation. Maybe you have to share a bedroom with three of your cousins for a year, and it's a little loud and crazy most of the time, but you can handle it. It's a tenable situation for you. For me, it would be untenable—I need my peace and quiet!
An untenable situation is NOT fine: you can't handle it, and you can't keep living your life in that situation. If you work 16 hours a day, 7 days a week, that's an untenable schedule, and an untenable situation. You're headed for a nervous breakdown.
That's mostly how we use the words "tenable" and "untenable:" to talk about plans, solutions, and situations. We can also apply these words to theories, positions, and other things.
A tenable theory is a good one: it makes sense, it holds together. But an untenable theory is a bad one: it quickly falls apart.
A tenable position is a secure one: it's comfortable, and you can stay in it for as long as you need to.
But an untenable position is a shaky, insecure one: it's uncomfortable, and you can't stay in it. You can't hold on. Something's gotta give. Here, Kip and Amy from Futurama are in quite literally an untenable position: they can't hold on.