In pictures, soldiers normally seen to have helmets, though helmets that only cover the top of their head, not the sides. Why not? They also don't seem to wear any kind of body armour, bullet-proof vests, etc. Was that because it wasn't available? Or too impractical?
Or did they wear more armour, that just isn't obvious in pictures? Did certain kinds of units wear more armour?
There were experiments with body armor during World War II (and even some during WW1), most famously including the Russian SN-42 steel engineer's breastplate. The SN-42 and similar steel cuirass designs were, broadly, protective against fragmentation and submachine gun fire at combat distances. Soviet soldiers reportedly found them beneficial in urban fighting, but heavy and uncomfortable. I am also aware of American experiments with aluminum-based plates, including aluminum/Doron fabric laminates, mostly as an avenue to improve aircrew protection over the ballistic nylon flak jackets. It would seem some of these designs could resist fragmentation & splinter threats similar to .30 carbine. However, I am not aware of these plates being fabricated into vests for air or ground crew issue.
None of these armor types were resistant to full-caliber rifle fire at combat distances. While it would certainly been possible, from a metallurgical perspective, to produce steel armor capable of stopping .30-06, 7.62x54R, etc, it would have been prohibitively heavy. American studies for the Variable Body Armor program during Vietnam concluded that the maximum useful density for infantry body armor was about 7-7.5 pounds per square foot; beyond this, it's just too heavy and fatigues the poor bloody infantry too much. For comparison, modern AR-500 armor steel is about 12 lb/sq.ft. There are modern armor steels that surpass AR-500, of course, but as far as I am aware those were mostly developed during vehicle armor research during the Cold War.
There was also research before and during the war on ballistic nylon armor, as used in flak jackets, but again - this stuff is good for fragmentation and possibly pistol ammunition. As far as I am aware, nobody issued it to ground troops, likely because early ballistic nylons were heavy, stiff, and hot - traits which are okay for bomber crews, but not so good for Private Snuffy the rifleman.
Sources:
Crouch, Ian G. The Science of Armour Materials. Duxford: Woodhead Publishing, 2017.
"Resume of Programs on Development of Body Armor Undertaken at Watertown Arsenal During World War II." Defense Technical Information Center, Accession Number : ADA954943. https://apps.dtic.mil/dtic/tr/fulltext/u2/a954943.pdf