Yes, it is absolutely possible. We live in a 1 bedroom flat (tiny hallway, tiny bathroom, tiny kitchen - no room for table, large lounge room and decent double bedroom) with our 10 month old and both my husband and I work from home. We are lucky, as the cot fits next to our bed but, if it didn't we could switch the lounge room and bedroom around.
In terms of baby 'stuff', much depends on how much you choose to buy them (or allow doting grandparents / friends to buy!). We sell things the baby has outgrown (e.g. jumperoo) to make room for anything new we wish to buy. We don't buy lots of toys and I keep the baby clothes in a holdall under our bed, as there is no room in our shared wardrobe for baby clothes, and no room for a chest of drawers or a 2nd wardrobe. We are also lucky that we only have two neighbours in our building, both of whom are happy for us to keep our pram in the shared hallway, so I don't have to carry it upstairs or store it inside the flat.
Empty boxes and paper keep our little one happy for hours, as does banging on pots and pans with a wooden spoon. We don't have any outside space, but have several parks nearby, and at least one walk a day in the fresh air does me and baby good.
The one thing I would suggest is a nappy bin - nappies can smell unimaginably bad, and I've found our nappybin keeps these smells encased, so the flat smells clean. The bins are small and cheap on Amazon, though the bags to line them are expensive. We wait for specials and stock up - on everything (bin bags, baby wash, nappy cream etc).
My parents managed to live in a very small caravan (not the huge Amercian-style ones made now) with me as a baby. Anything is possible. Easy, no, but possible, yes.