The calculated distance (air line) between Modi‘in Makkabbim Re‘ut and Eilat is approximately 161 miles respectively 260 kilometers.
By car or train, the actual journey to Eilat is certainly longer, as only the direct route (as the crow flies) between Modi‘in Makkabbim Re‘ut and Eilat has been calculated here.
Air line (approximately)
260 kilometers
140 nautical miles