Provided that the restriction against crossing any lines means only that none of the utility lines can cross, then the puzzle can be solved without making any assumptions about lines passing over one another, or being underground, or the like.
Here is one way of doing it:
For ease of reference, mark the houses going from top to bottom as A, B and C. Number the squares representing utilities from left to right as 1, 2, and 3.
Utility 1 can be connected to each of the houses my drawing lines to the left side of A, B and C.
Utility 2 can be connected to House A by drawing a curving line which begins at the bottom of Square 2 and then arcs around on the outside of the three lines extending from Square 1. Utility 2 can be connected to House C by a simple straight line. Utility 2 and House B can be connected by drawing a line which loops around from the right end of the top of Square B to the right side of House B.
Utility 3 can be connected to Houses A and B by drawing lines from Square 3 to the left sides of Houses A and B.
This leaves only the problem of connecting Utility 3 and House C
If the rule against crossing lines refers only to crossing utility lines, one can draw a line from the bottom of Square 3 and loop it up and through Square 2 and then directly into House C. Alternatively, one can draw a line from the bottom of Square 3 and overlay it on dividing line between Square 2 and Square 3. Arguably this line and the dividing line do not “cross”.