CL65 FLT CREW TAXIED ONTO ACTIVE RWY AT CLE DURING PERIOD OF RWY CLOSURE.

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I TAXIED THE ACFT ONTO AN ACTIVE RWY (RWY 24C) AT CLE; CAUSING ANOTHER ACFT TO EXECUTE A GAR. THE INVOLVED PARTIES INCLUDE THE FLT CREW; CLEVELAND GND CTL; AND TWR. SEVERAL FACTORS CONTRIBUTED TO THIS OCCURRENCE; BUT ULTIMATELY THE FAULT IS MY OWN. WE CONTACTED GND AT SPOT X AND WERE GIVEN THE TAXI INSTRUCTIONS TO TAKE TXWY J TO TXWY A TO TXWY R AND HOLD SHORT OF RWY 24C AT TXWY B. THESE INSTRUCTIONS WERE NOT MISUNDERSTOOD AND VERIFIED VIA THE 10-9 PAGES. ADJACENT TO RWY 24C; RWY 24L IS CLOSED AND CURRENTLY UNDER CONSTRUCTION. THE CONSTRUCTION CONTRIBUTED TO A LOSS OF POSITIONAL AWARENESS. I NEVER SAW THE HOLD SHORT MARKINGS FOR RWY 24C; AND DIDN'T REALIZE THAT I WAS TAXIING ONTO AN ACTIVE RWY UNTIL I HAD ALREADY DONE SO. ALSO; APCHING RWY 24L; GND CTL SWITCHED US OVER TO TWR; WHO GAVE THE CLRNC TO HOLD SHORT OF RWY 24C AT RWY 6L. THIS CREATED SOME CONFUSION AS IT DID NOT MAKE ANY SENSE. I UNDERSTOOD THE CLRNC TO MEAN HOLD SHORT OF RWY 24C AND RWY 6C; WHICH WAS THE INTENDED MESSAGE. THE FO WAS ATTEMPTING TO VERIFY TWR'S INSTRUCTIONS TO US BY EXAMINING HIS CHARTS AND; AS A RESULT; WE HAD 1 LESS PAIR OF EYES OUTSIDE OF THE ACFT. I REALIZED MY MISTAKE ALMOST IMMEDIATELY AFTER IT OCCURRED. UNFORTUNATELY; IT WAS TOO LATE AT THAT POINT TO SIMPLY STOP AND HOLD SHORT OF THE RWY. WE IMMEDIATELY INFORMED ATC OF THE ERROR AND EXPEDITED OUR TAXI OFF OF THE RWY.