Put a clipping mask on the layer that contains the shapes.
Yes, the shapes will still be there after reopening. Normally that's how you would want them, because then you can change them easily.
But if you really want to cut them, there's no easy way to do so in Illustrator when strokes are concerned. So masking would basically be the way to go.
You can cut them (albeit with about twice the work required by drawing programs which provide a decent cutting tool):
1. Line Tool: ShiftDrag to draw a vertical line along the page edge, crossing all the paths you want to cut.
2. Black pointer: Select the line you drew and the lines you want to cut.
3. Shape Builder Tool: AltDrag across the unwanted portions.
4. Select and delete the vertical line.
Always keep (a copy of) artwork that you may need again (even if you are completely convinced it will never happen), on its own layer or whatever.