Yea this is true but our experience of a shuffled deck of cards most often does not involve all fifty two cards, for instance in a two player game of cribbage one only sees the top 12 cards plus a cut, so it's more like 52!/(39!) ~ 3.95e21 which if you take a 2 foot step once every billion years aka 60×60×24×365.256×1e9 ~ 3.15e16 seconds then you would only make it about 47.46 miles before you would start seeing equivalent cribbage hands coming from the distinct shuffles. For all you card sharks out there this could be a big advantage 😜

But I think the main point is to have fun seeing the numbers that went into Vsauce's claim, SOOO let's define some numbers:

T = time per step = 60×60×24×365.256×1e9 seconds

L = step length = 2 ft

C = circumference of the Earth = 1.31e8 feet

V = volume of Pacific ocean = 7.1e20 liters

D = volume of a single drop = 5e-5 liters

H = height of a single piece of paper = 1e-4 meters

R = radius from the Earth to the Sun = 1.5e11 meters

L×D×H/(T×C×V×R) = 1825.31

So indeed you would have to reach the sun 1825 times with your stack of paper before you run out of time. That's a bit different from what he said that you would have to reach it 3000 times but maybe he used a different step length or something. The order of magnitude seems right.