All the same race is pretty easy. I think four out of five parties I play in are almost all human anyway. I played in one all-dwarf campaign for a while too. That's pretty much standard D&D with a little bit stronger hook for getting and keeping the party together. (All the dwarves were from the same clan, etc. An all human party could be from the same tribe or village, etc.)
All the same class could work too but would probably work best if multiclassing or gestalts were allowed. A party with a fighter/cleric, a fighter, a fighter/rogue, and a fighter/wizard could work pretty well (though the fighter/cleric and fighter wizard would probably have most of their levels in the other class). Similarly, a party with a fighter/mage (Eldritch Knight), Arcane Trickster, Mystic Theurge, and straight up arcanist could work quite well. Without multiclassing and staying close to the traditional D&D adventures, an all druid party (shapechanger, offensive spellcasting focussed, healing focussed, summoning focussed, and skills focussed druids) would work well (as long as they had appropriate adventures), as would an all cleric (battle cleric, healing/support cleric, offensive spellcasting focussed cleric, and an archer cleric) party. An all paladin, all ranger, all bard, or all monk party could also work though the limited magic would make it more of a challenge. An all rogue, all fighter, or all barbarian party would pose more of a challenge because the lack of magic would dramatically change the way they had to face some challenges.