I'm trying to create a box with a lot of corners in it but it would be helpful if I could make the corners without having a break in the lines. Is there a way to draw lines continuously without any breaks at all?
Use the pen tool, it does what you want as long as you click the corner point and do not attempt to drag the segment. Line tool can not make a polyline just individual lines.
Though, its also no problem to click drag multiple times and join the lines with shapebuilder or ctrl + j afterward. It may seem like a extra step but given you often need to compose lines and shapes separately to do proper T-junctions for example it's not exactky as wasteful as it seems
Actually it can be done with the line segment tool. I use Illustrator CC 2018. Draw a line anywhere on the page. Use menu at top right to verify "Shapes" is checked. This will bring up a little shape option in the toolbar. "Show shape widgets" is the default. Click it to "Hide shape widgets". Then you can use line segment tool (NOT PEN) to draw multi-line shapes. You will still need to join the lines but it is much quicker than the pen tool.