If they don't know you or have anything to ask you of, most people in Russia won't even look at you when you pass them by. Oddly enough, it's entirely normal - for a Russian person.
For some reason, eye contact is deprecated in Russia. The only good reason to look at another person is to tell them "I saw that", "You're out of your place", "I need something from you but can't come up and ask for it" or "I like you but can't come up and tell you that", with the difference articulated through the language of the facial muscles. If you try to make eye contact, the person might look back at you - and turn away a second later. Most people will consider odd looking at someone for long time; some will call you a creep (at least mentally) and move somewhere else.
Russian people prefer to not deal with strangers, and this trait dictates most of their behavior. If you look sick on the street, a rare person will come up and ask if you're alright, let alone call the ambulance. The better you look, though, the more people you'll attract by looking sick: if you're a hobo, you're most likely going to die if what you have is lethal; if you're a dandy professor, you have much higher chances of survival. This, however, only applies when you've been sich before people saw you: if you suddenly and visibly turn sick, people around you will come and help. Make sure to attract attention by loud yells or breaking something that breaks loudly if you're ever seriously sick in Russia.
Still, if you come up to people and, say, ask directions, they will tell where to go if they know. Asking directions in general is a good way to attract a person's attention in Russia, but try to cheat your way into their zone with that: they will resent you more than they would if you'd come and ask what you wanted to directly. It's common to see people ask for a change on the street, whether it's for a ride home or another bottle of vodka (see me explaining to Elisza exactly how much Russians drink), and it's as common to refuse them - most don't get offended and come up to another person a dozen meters away.
If you live in the same podyezd*, you're generally expected to at least greet when you see each other; however, it's not mandatory, and one can live for years without knowing their neighbours' names or even the apartment they live in. If you know one of your neighbours, even if they live on a different floor, it's not fruitless to ask them for help: refusing help is seen as very rude by some, even if the ask is very big. Since the Soviet times, lending each other immediate kitchen needs - sugar and salt are most popular - is something many people will do happily, as well as lend you chairs or plates for a big occasion. The latter is in rare practice these days because most people have enough place to sit or to put food on, even if it means overstuffing your apartment or your house. Serving others' needs is seen as a necessity rather than a choice and is often more important than one's well-being.
If you've found an older person who holds you dear - perhaps, your love interest's mother or grandmother - you're to be treated with much of kindness and tenderness. Such people will care about your life a bit too much, because they'd like to make sure you have everything good. If you don't, they'll go out of their way to help you a bit too much as well, but always with best intentions. It's an old Russian tradition to treat your guests with tea/coffee or some food (it's fine to refuse either, even if they insist), so such people will always heat up a big bowl or plate of what they've got in their refrigerators - which, if you're lucky, is a well-prepared bortsch - sometimes even without asking you. It stems from the fact that good food used to be rare in the Soviet Union, so treating a guest with something good means that you care about them quite a bit; old women - called babushka ([BAH-bush-kah], literally "grandmother"; plural "babushki", [BAH-bush-kih]) - are especially prone to it.
Russian people's deeper feelings are generally closed off for strangers, only opening up to people they trust. They might tell you about their life in general even if they don't know you, but the rest is under strict protection, so dare to learn others quickly at your own risk. Two people who spend their time in proximity - neighbours by podyezd or co-workers - might get along just fine on a superficial level and never let others into their inner world because in Russia, expressing your emotions is considered a weakness; only women are tolerated being emotional because they're still perceived as inferior to men by the general population. Expressing one's deeper feelings is only tolerated by the public when the exchange occurs between people who are perceived to must be having such a connection, like husbands and wives or, far more rare, parents and children.
Honesty and blunt truth aren't appreciated in Russia. It is seen as being polite and a good boy/girl to not say the ugly truth, no matter how important it is, and as being rude otherwise; some people - particularly older women - will do all kinds of nasty tricks if you've offended them by talking straight, while others might simply start ignoring or passively despising you afterwards. The level of comfort varies, of course, but if you're looking to be liked by Russians, never speak the truth, and superficially, you'll get along just fine. Because of that and the previous traits, people aren't willing to talk about what really bothers them, the more stress they experience the less. Thus, it's quite common to see the folks from one podyezd - sometimes joined by people from other ones - to gather together and laugh at some stupid shit while drinking beer or trying hard to control their children.
Here, I must make a note: I've only experienced living in one podyezd of one house and haven't been anywhere else for long, so I may be making assumptions which are wrong for other parts of the vast and different country.
To recap:
- If people aren't reacting to you - don't mind and keep asking;
- Don't look at strangers for long unless you're getting the signals of interest;
- Some people are deeply reluctant to help while others can go out of their way to do so; there are no outside signs beyond the generally cheerful and happy attitude;
- If you mean to live in Russia and you care about others' opinion of you, take note of maintaining good relationships with your podyezd or yard's most vocal people, for they're the ones who can praise you or ruin your reputation in one evening;
- Even if they care your life a bit too much, older people do it out of best of intentions; it often stems from the fact that, as they retire, they lose things in their life that made it exciting, and so they grow bored - so, when they meet someone to talk to and care for, they will jump at the opportunity without even noticing it;
- Don't try to reach out to people who aren't willing to talk about their feelings unless you're getting the signals of wanting but being too shy or too scared to admit being weak, in which case do press softly and, when they start talking, proceed as you would with such a person in other countries;
- If you care about you being honest and/or open with others, be prepared for opposition and don't listen to those who'd tell you to stop it - there'd be plenty;
* for the definition of "podyezd", see this comment