This question has received a lot of upvotes, but no answers, probably, I suspect, because it's a bit difficult to provide a one-size-fits-all response, and the concerns you express are also distinctly modern ones, for reasons that I hope will become clear in what follows. Indeed, what follows can be summarized by saying: (a) the ancient world was overal a lot poorer in material terms than our modern one and; (b) it is extremely unlikely that in the ancient world homes would be left entirely unoccupied for any stretch of time.

You refer to your wallet and depending on the time period, someone leaving the house would indeed bring money – i.e. coins, first introduced in the seventh century BC in the form of large-scale currency – with them, stored in a purse of some kind, if they intended to spend anything. An example would be a Roman – or more likely, a number of Romans (e.g. the mistress of the house and a slave or two) – going to the market or a shop to buy something). In a barter economy, this requires more planning because of how cumbersome the exchange of goods can become (even if the principles of trade are essentially the same). For more, you might want to check out, for example, the papers in W.V. Harris (ed.), The Monetary Systems of the Greeks and Romans (2008).

When we leave our homes today, we usually lock the door, usually when there's no one at home. A very brief overview of the development of more or less secure mechanisms to lock doors is useful here (there's a fun overview in Brian Fagan's edited volume The Seventy Great Inventions of the Ancient World, published in 2004). Doors with hinges are a development of the Iron Age -- before then, doors turned on pivots set into lintel and threshold, and could be barred from the inside only. Latch-lifters were developed in Greece and China in the earlier part of the first millennium BC. Tumbler locks, using wooden pegs, emerged later in the Greco-Roman world as well as China. Locks were mostly used to secure areas that were left unattended or where it was important to slow thieves down (e.g. the store room of a temple); seals were used for this purpose, too, from the Bronze Age onwards, but they did little to stop someone from accessing whatever was sealed (e.g. Tutankhamun's tomb).

As you might suspect, the relatively late development of means to effectively lock doors means that this wasn't considered much of a problem before the Iron Age, and even then, there are different factors at play then than there are today. In ancient times, you would generally try to avoid situations where no one would be left behind, especially since most houses were fairly easy to break into (both wattle-and-daub and mudbrick are susceptible to being holed using metal implements). There would usually be someone at home at all times or, depending on the type of building in question, guards would be posted at the doors. In the Greek and Roman world, women were mostly left at home while men went out; in the Bronze Age, women had more freedom, but even then, situations where a house was left completely unattended would probably have been fairly rare. In the Classical world, most families owned at least one slave, including smallholders.

It's a feature of the modern era that most residences are occupied by single people, couples, or nuclear families (couple + their young children). However, in ancient times, households tended to be bigger, with at least some of the children staying with their parents into adulthood: e.g. in Classical Athens, sons of at least affluent families, of which we know more, usually didn't marry until they were around 30 years old. Even e.g. craftsmen could rely on their (older) children, slaves, apprentices, or possibly even landless labourers who worked in their employ (i.e. the poorest of the poor), to hold the fort in their absence if necessary.

As regards to what people would take with them when they left the home, the answer is "usually very little", because the ancient world was, compared to our own, a lot poorer in material terms. But it depends on the context. Archaic Greek vase-painting coupled with e.g. the evidence from the Homeric epics (which mostly refer to Homer's own time, i.e. around 700 BC at the earliest: for more on this, see my article here), suggests that no able-bodied high-ranking man would leave his house without at least a spear at his side; "bearing arms" was later considered not done and replaced, before the end of the Archaic period (ca. 500 BC), with some men carrying wooden staves instead (see Hans van Wees's paper, "Greeks bearing arms", in Archaic Greece: New Approaches and New Evidence, the volume he edited with Nick Fisher, published in 1998). If you had to travel far, you'd put on a hat and bring supplies (or have a slave and/or pack animal carry them for you), and so on.

I hope this goes some way to answer your question; feel free, of course, to post follow-up questions.