You are really in a dilemma. It happens now and then for most people who wear contact lenses. Normally speaking, wearing contact lenses is safe if you wear them for less than 10 hours per day. It is very dangerous and bad to wear them when sleeping. So we surely need to take off the lenses and store them somewhere if we have no cases by hand. I once saw someone introduced that if you have no contact lens solution, you can use eye drops to clean and store them. Distilled water or saline solution is acceptable too. But water directly from tub is not good. If you don't have cases either, find two bottle caps to keep them. It is important for you to remember which one is for which eye. After all, it is only an alternative option if you happen to have this headache. It is much better if you can take solution and cases with you.
There are some substitutions that can be used as contact solution if you don't have any in a emergency, such as saline solution, distilled water, and salt water.
Saline solution is the best alternative for contact solution, but don't use it often. Distilled water is clean so it won't dirt your contact lenses much, but you still holds to chance of being infected. If you can get the above alternatives, there is a more easy way to store your lenses. Make some salt water mixing with one teaspoon of salt and six to eight ounces of water. Salt water can kill most of the bacteria but not all of them. Remember to clean the contact lenses next time you wear.
Oh, I also have such a same experience. If you do not have contacts solution and contact lenses case, you can use the alternative distilled water to store your contacts. Distilled water should not carry acanthamoeba keratitis, although it is still possible to get an infection if the water is not sterile. Although distilled water is not a foolproof option for storing contact lenses, it is safer than regular tap water. Or you can use bottle cap with distilled water to store your contact lenses.
If you have to store the contact lenses in emergency but with no contact lenses cases and contact solution by the hand, you could put the contact lenses in the case with purified water which may have little invisible bacterium. Then you have to take it out and put them in the contact solution to make them clear the lenses. If you could have the contact solution, you'd better use it to clear your contact lenses.