I suspect this is a hardware problem. From your Boot Repair output:
ERROR: asr: reading /dev/sda[Input/output error]
ERROR: ddf1: reading /dev/sda[Input/output error]
ERROR: ddf1: reading /dev/sda[Input/output error]
ERROR: hpt37x: reading /dev/sda[Input/output error]
ERROR: hpt45x: reading /dev/sda[Input/output error]
ERROR: isw: reading /dev/sda[Input/output error]
ERROR: isw: reading /dev/sda[Input/output error]
ERROR: isw: reading /dev/sda[Input/output error]
ERROR: jmicron: reading /dev/sda[Input/output error]
ERROR: lsi: reading /dev/sda[Input/output error]
ERROR: nvidia: reading /dev/sda[Input/output error]
ERROR: pdc: reading /dev/sda[Input/output error]
ERROR: pdc: reading /dev/sda[Input/output error]
ERROR: pdc: reading /dev/sda[Input/output error]
ERROR: pdc: reading /dev/sda[Input/output error]
ERROR: pdc: reading /dev/sda[Input/output error]
ERROR: pdc: reading /dev/sda[Input/output error]
It goes on like this for quite a while. Input/output errors are usually (but not always) indicative of failing hardware. This diagnosis is also consistent with the fact that Boot Repair thinks your /dev/sda
doesn't have a valid partition table, but the OS thinks it does have a partition table. (Hypothetically, if the hard disk has become unreliable, the OS might have read the partition table and then Boot Repair might have been unable to do so a short time later. It could also be that the first sector is bad but the OS read the backup GPT data from the end of the disk. If Boot Repair doesn't look there, then this would explain why it thinks the disk has no partition table.)
You may want to run SMART tools on the disk to test my hypothesis. If the SMART results look bad, then the disk is having problems, and it may be time to replace it. (See shortly for a caveat, though.) Note that disks can sometimes fail while SMART seems to think they're OK, so a clean result on SMART doesn't necessarily mean the disk is really OK. On occasion, SMART will complain about read errors, but attempting to write to the affected sectors will cause the disk to remap the sectors, and everything will then work OK for a while. Despite these caveats, running SMART on the disk is worthwhile -- its results are usually helpful.
There can be causes of hardware failure other than a bad disk. A bad cable, or even just one that's not properly plugged in, is one example. On occasion, the disk controller circuitry on a motherboard will fail, in which case you'll need to add a plug-in disk controller card (if this is possible) or replace the motherboard.
If I'm right that this is failing hardware, you should stop using the computer immediately! Buy a replacement hard disk (if that's the failing component) and, once it's physically in your possession, hook it up to the computer along with the failing one. You can then use various recovery tools and procedures to copy the old disk's contents to the new one. Depending on how bad the failing disk is, you might be able to copy data with normal backup tools; or you might need to use something like ddrescue
to recover data from failing hard disk sectors.