If you are talking about texture maps,then it depends very much on what your final output size is going to be.

You need to work out approximately what the largest size that any portion of the texture will appear in your final pic or if an animation screen size . This is obviously simpler for a final still image than an animation since in a still the whole character will be visible (so final output size x 2 ).But for animation if you are texturing a face that is going to be seen full-screen at any given point in the animation, then you want to make sure that your texture maps have enough resolution on the face to look good if it fills the screen. If the camera zooms into the face, and there is a close-up shot of say the characters nose, then you are going to have to make certain that the texture maps for the face are large so that they will stand up even when most of the time only the full character is viewed full-screen.

Once you have calculated the largest size that any portion of the image map will appear on the screen, you can work out an appropriate size by then multiplying that size by two, to give you a total map size

If the animation was for television at a resolution of 720x576 pixels then in order for it not to appear blurry you need at least 1440 pixels so a 2k map would be okay but if you plan to zoom in to the nose and the nose only takes up a tiny part of the texture then your map would have to be so huge it would not be possible or practical so you might want to texture the head on a separate map.
If you have no close ups or a still image you need to calculate the smallest thing that might appear full screen.. lets say the head for example.

If your head takes up one third of your texture map then that one third of the map (the head) will be appearing at a size of 720x576 pixels, and so therefore the image map for that area of the head on the map has to be at least twice that size (1440). In order to be sure that the image map stays sharp for close-up, you will need to firstly multiply 720 by three (because your head takes up one third of the total area)= 2160, to get the total size of the image maps for the entire character, and then double it to get a safe size to make sure that the quality stays acceptable. so now the size of your image maps would need to be 2160 x 2 = 4320 so your texture maps would be 4320 x 4320 pixels since UV maps are square or 4k .