I think "significant about" might be overstating it. Most Koreans can read names in 한자 and some language basics (colors, descriptive adjectives, etc) but I don't think the average Korean wouldn't be able to pick up a text written in Hanja and understand it.

I had an interesting experience with this a few years back. My grandmother was Korean and left in 1953 for the states. I moved to Korea in 2011 and needed to get a visa. There is a visa for people who are descendants of a native Korean, but you have to prove your lineage. My grandmother had a Korean passport from 1953 that was written in 한자 and English, but not 한글. I brought it to a city office with it to get my family tree, but the mid-20s worker could speak English and Korean, but couldn't read 한자. She got her 45ish boss who couldn't speak English but knew more 한자. The middle aged woman ended up getting the oldest man in the building (90ish?) to come translate the passport because he was the only one who could read enough 한자.