I’ll note that Nalgene bottle have a seemingly great design with a V-shaped groove within the lid that pinches down on the top lip of the bottle. Seems like a great design and someone probably patented it. But that leaves little micro-passages from inside to outside with no ability to hold back pressure, oils leak out with minimal pressure – 0.1 atmospheres (e.g. going from sea level to 4,000) and less.
Water stays inside because the very high surface tensions in small passages resists such modest pressures. As passage size decrease, surface tension takes off, required very high pressure to overcome,
But what holds in oil in the pressure encounters in, say, an auto engine? Gaskets. Squashed between two surfaces. That’s what PEET soda bottles do. The lid has a cushy gasket material inside of the cap – it holds back 150 psi (until the body of the bottle rips apart – I suggest wearing good hearing protection).
As practical, free solutions, I’ve used:
The smallest, clear plastic soda bottles (Coke, 7-Up, etc) like an 8-ounce (237-ml) one for olive oil. Which I bring for a cold lunch of reconstituted tabouli and adding those dense calories to other dishes that didn’t call for it. I’ve actually never had a problem with more crinkly disposable water bottles, but I feel safer with the name-brand soda or upper-end less crinkly water bottles.
And, outside of every convenience store, there is no end of “5-energy” 2-oz plastic bottles which are quite sturdy, dishwasher safe IME, and free.

There are also lots of “nips” – those little 50-ml bottles of cheap vodka, Fireball Whiskey, etc. The metal caps are crap (but fine for some powders or spices) while a few with a plastic cap hold pressure just fine. 
Also free. Reduce, Reuse, Recycle!