The key to how close is to keep an eye on your mêlée skill. It will be unusable too far away. Approach the MOB, as others have said, from behind (in Lotro, that's 180°), and when your skill becomes usable, IE, it's icon changes on your skill bar, use it without getting any closer.
Things that make you more noticeable include being in front, moving, and level difference, you can get much closer to lower level things than higher (without items that increase your stealth level) without being noticed.
Hide in Plain Sight makes you completely undetectable for 10 seconds to MOBs, so if you do need to just get past something and don't mind burning it's long cooldown, that's an option too.
Note that in instances, many things are triggered based upon player position irrespective of stealth, so don't expect fights that trigger based on proximity to let you sneak in undetected.
Link to our community LOTRO store google spreadsheet pricelist and conversion rates, please contribute too!: https://goo.gl/wxPqCm