Though 'bij' often means 'close to' or 'near' (something), this meaning really isn't recognized in this context.
What you can say is that the children live 'near us', for example:
- "De kinderen wonen in de buurt (van ons)." = "The children live near/close (to us)."
Literally: "The children live in the vicinity."
- "De kinderen wonen dichtbij/vlakbij ons." = "The children live near/close to us."
When we're talking about 'wonen bij', however, what is conveyed is that the children live with someone (in this case 'us'). It is a combination of a verb with a preposition that expresses a particular meaning or activity: people living together or someone living in the house with someone else, the latter person who owns the house. :)
- "Ik woon bij mijn tante." = "I live with my aunt." or "I live at my aunt's."
- "Hij woont bij zijn vriendin." = "He lives with his girlfriend."
- "Hij woont bij zijn vriendin (in)." = "He lives in the house of his girlfriend, together with his girlfriend."
I saw an explanation of this that seems to capture it. The idea is that 'bij' means alongside, so from a Dutch native speaker's point of view the guy has a glass of water alongside his dinner. I was wondering how this would work out if you were trying to express the idea that the glass of water was part of your meal, but it just occurred to me that it might not make sense to use the word 'with' if you don't see the dinner and the glass of water as two separate things. If a Dutch native speaker did not see the dinner and the glass of water as two separate things, I don't think that the Dutch native speaker would use 'met' or 'bij'. Instead the Dutch native speaker would say something like: "Hij drinkt een glasje water als een deel van zijn avondeten".
I think the reason is that if you don't write out "zesentwintig", then you're not showing that you've actually learned how to say or write that number in Dutch! (Also, just generally speaking, you wouldn't switch between the short and long forms when performing a proper translation.)