This is probably "Rogue Ship" (1950) by A.E. van Vogt.
Averill Hewitt discovers that the sun will become variable and destroy life on Earth, and in 2072 builds a ship to send colonists to Alpha Centauri to save humanity.
The story opens six years later as the ship, Hope of Man, is sighted approaching Earth when it should be nearly at Alpha Centauri. No communication is made with the ship, and its physical properties are greatly changed - the best drill in the world can't penetrate the hull, the ship itself plows several hundred kilometres through the Earth completely undamaged and unslowed, and the air pressure inside is measured at 973 bar.
Inside the ship, things are greatly distorted (though the ship appears normal from the outside). Light is strange and dimensions are compressed or expanded; all the people are strangely flattened in the direction of flight, though the factor doesn't match the expected Lorentz contraction. (All the expected crew and passengers are present, they simply appear frozen.) The captain is found at his desk in the middle of writing an entry in the ship's logbook.
It becomes clear that the ship is experiencing a time ratio of 973:1 relative to Earth, travelling at a rate of over 177,000 miles per second, while nevertheless appearing at a relatively sedate 1,000 miles/hour in the solar system.
Hewitt goes aboard Hope of Man to pass instructions to the captain and scientist in charge, to get them out of the strange situation they are in:
...without further incident he delivered one copy of the letter to Grayson and another to Tellier. He was greatly stimulated to notice that Grayson had finished writing the word "happened" during his absence. He could see no change, however, in the position of Tellier.
The story was first published in Super Science Stories, March 1950 and it can be read at the Internet Archive. If you read it in an anthology in the 1970s it was probably Tomorrow and Tomorrow (1974), ed. Damon Knight.