The 'Middle-Way' does not require giving up all worldly possessions. Instead, the Middle-Way states to not engage in sensual pleasures.
A person can have basic worldly possessions ('requisites'), such as food, housing, clothing, medicine, etc, and still practise the Middle-Way.
In short, practising the Middle-Way does not require being a monk or nun.
The 'Middle-Way' was described in the 1st Sermon as the way of life that does not rely on impermanent & unsatisfying sensual pleasures for happiness nor subjects oneself to & attempts to become impervious to pain.
The Middle-Way culminates in the four blissful meditations (jhana) and ultimately Nirvana, which is permanent peace & freedom. The happiness of the Middle Way is far superior to the happiness of sensual & worldly pleasures. Therefore, there is no contradiction.
People are attached to sensual pleasures, which generates suffering when those sensual pleasures cease &/or are lost. Sensual pleasures also create enslaving tormenting addictions. Further, sensual pleasures cannot bring true permanent happiness, which is why people get bored of sensual pleasures and continually search for & acquire new sensual pleasures.
Therefore, from a Buddhist perspective, reliance on sensual pleasures cannot be bring lasting true permanent happiness. The scriptures state:
The Blessed One has said that sensual pleasures are of little satisfaction, much stress, much despair & greater drawbacks. The Blessed One has compared sensual pleasures to a chain of bones...to a lump of flesh... a grass torch... a pit of glowing embers... a dream... borrowed goods... the fruits of a tree... a butcher's ax and chopping block... swords and spears... a snake's head: of much stress, much despair & greater drawbacks."
> *And what may be said to be subject to aging... illness... death... sorrow... defilement? Spouses & children... men & women slaves...
> goats & sheep... fowl & pigs... elephants, cattle, horses, & mares...
> gold & silver are subject to aging... illness... death...
> sorrow... defilement. Subject to aging... illness... death...
> sorrow... defilement are these acquisitions, and one who is tied to
> them, infatuated with them, who has totally fallen for them, being
> subject to birth, seeks what is likewise subject to aging...
> illness... death... sorrow... defilement*
>
><a href="http://www.accesstoinsight.org/tipitaka/mn/mn.026.than.html">*Ariyapariyesana Sutta: The Noble Search*</a>