Most people are giving you good advice on how to make a good sauce for cheap, but I hear you on the low-equipment and storage issue.

I'm going to assume you have access to a microwave or portable stove top/burner or other heat source, a glass bowl or cooking pot, something to stir with, and maybe a tiny little space in a fridge.

• Dried herbs and spices are your flavour friends and are shelf-stable. For pasta sauces you can't go wrong with the traditional Italian herbs basil, oregano, thyme, rosemary. Some places sell a premix "Italian seasoning", I prefer to mix and choose myself to help keep some variety. You do what is best for you.
• Bottled vinegars help with flavour balancing/boosting, if it's bland but already contains salt and fat, a tiny bit of vinegar will help. If it's too sweet, adding a little more vinegar will help that too. And they are shelf stable. For pasta sauces a red wine or balsamic or regular white vinegar is fine. Try to get vinegar without added sugar. Bottled lemon juice is also OK, but crap compared to fresh lemons. Fresh lemons can keep for a while in a fridge if you have space, and you don't need much to get the effect.
• Some kind of sugar, for the occasions where you get really sour unripe tomato or you put too much vinegar in. Or if you get some bitter vegetables/flavour. White sugar is fine, I like light brown better. Honey works too in even smaller quantities. All raw sugars are shelf-stable.
• A bottle of vegetable oil (or olive oil, especially for italian/greek). Again, shelf-stable. Fat boosts flavour (and calories, be careful with it).
• Bullion or Powdered Stock, also shelf-stable. Adds salt + savory flavours. Other shelf-stable options for salty + umami are soy sauce but it isn't very Italian. Dried mushrooms work for savory flavours but might be hard to find or expensive or not the right kind. I have a small container of dried porcini mushrooms for European foods, and shiitake for Asian foods.
• Bottled pasta sauce, try for no added sugar, oil or salt, because you will add your own and they are easy to over-do. Premade pasta sauce is often the cheapest shelf-stable option including multiple vegetables and has some pre-balancing of flavours already done, but it can be shit quality ingredients. It is precooked just needs to be heated through with whatever heat source you have. If you hate the flavour of it, try to add a little of each of the other items while heating until it tastes better. Understanding balancing the "5 flavours" fir your own tastes is key to mastery. If you only eat/heat part of the sauce, the rest of the container needs to go in the fridge or freezer.
• Passata or bottled/tinned tomatoes. This is the often higher quality and cheap option, doesn't have the advantage of other veg for nutrition, and it's slower to make because you need to heat it through for long enough to evaporate some water off to make it more flavourful. Again, try to avoid added sugar/fat/salt. Otherwise it has the same instructions as for pre-bottled sauce, heat, mix flavours, eat. If you don't use all of it, again, it needs to go in the fridge or freezer.
• Some tomato paste is close to shelf-stable, others need the fridge. Check the packaging but this is a highly recommended small-size way to get more intense tomato flavour.
• Other shelf-stable additions for pasta: Whole black pepper in a grinder is the best, but any works, or red chilli flakes, for some heat. Dried/fried onion flakes, dried chives, dried garlic flakes / salt... whatever you like best.

If you mix any or all of the above to your taste, heat it through and for long enough without a lid to thicken the sauce by evaporation, you can get some extremely tasty food. And the only things you will need to store in a fridge are the leftover end-result or tomato-based or rehydrated ingredients.

But, you will experience poor nutrition relying only on shelf-stable foods in longer situations. If you can boost that with a variety of frozen or fresh veg, you will be doing yourself a huge favour. You also need to consider protein sources. Meats nd cheese usually need the fridge, but beans and lentils in the right balance can work for no-fridge situations too.

I unfortunately have a lot of experience with this issue, so if there's anything else I can suggest solutions for, let me know.